Solve 3z=-1/z

$$3 z = - \frac { 1 } { z }$$

Solve for z

$$z=-\frac{\sqrt{3}i}{3}\approx -0-0.577350269i$$

$$z=\frac{\sqrt{3}i}{3}\approx 0.577350269i$$

Steps by Finding Square Root

Variable z cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by z.

$$3zz=-1$$

Multiply z and z to get z^{2}.

$$3z^{2}=-1$$

Divide both sides by 3.

$$z^{2}=-\frac{1}{3}$$

The equation is now solved.

$$z=\frac{\sqrt{3}i}{3} z=-\frac{\sqrt{3}i}{3}$$

Steps Using the Quadratic Formula

Variable z cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by z.

$$3zz=-1$$

Multiply z and z to get z^{2}.

$$3z^{2}=-1$$

Add 1 to both sides.

$$3z^{2}+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 0 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{0±\sqrt{0^{2}-4\times 3}}{2\times 3}$$

Square 0.

$$z=\frac{0±\sqrt{-4\times 3}}{2\times 3}$$

Multiply -4 times 3.

$$z=\frac{0±\sqrt{-12}}{2\times 3}$$

Take the square root of -12.

$$z=\frac{0±2\sqrt{3}i}{2\times 3}$$

Multiply 2 times 3.

$$z=\frac{0±2\sqrt{3}i}{6}$$

Now solve the equation z=\frac{0±2\sqrt{3}i}{6} when ± is plus.

$$z=\frac{\sqrt{3}i}{3}$$

Now solve the equation z=\frac{0±2\sqrt{3}i}{6} when ± is minus.

$$z=-\frac{\sqrt{3}i}{3}$$

The equation is now solved.

$$z=\frac{\sqrt{3}i}{3} z=-\frac{\sqrt{3}i}{3}$$